An indefinite concave-convex equation under a Neumann boundary condition I

TL;DR

This paper studies an indefinite nonlinear elliptic equation with Neumann boundary conditions, proving the existence of two non-negative solutions for small parameters, analyzing their asymptotic behavior, and suggesting a loop structure in the solution set.

Contribution

It establishes the existence and asymptotic profiles of multiple solutions for an indefinite concave-convex problem under Neumann conditions, including bifurcation analysis.

Findings

Existence of two non-negative solutions for small |λ|

Asymptotic profiles of solutions as λ approaches zero

Potential loop structure in the solution set

Abstract

We investigate the problem $$-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } \Omega, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial \Omega, \leqno{(P_\lambda)} $$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $\lambda \in \mathbb{R}$, and $a,b \in C^\alpha(\overline{\Omega})$ with $0<\alpha<1$. Under some indefinite type conditions on $a$ and $b$ we prove the existence of two nontrivial non-negative solutions for $|\lambda|$ small. We characterize then the asymptotic profiles of these solutions as $\lambda \to 0$, which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of $(P_\lambda)$.